Subgroups — C₂×#13114991 #15863315

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88048	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718637	C₂	1	2	yes	maximal
5	#13114991	C₂	1	8	yes	maximal, direct product factor
6	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#6101	—	2	4	no	—
11	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	—	2	4	no	—
14	C₂×C₁₃×#106	C₂²	1	4	yes	—
15	C₂×#88057	—	2	4	no	—
16	C₂×#88057	C₂²	1	4	yes	—
17	C₂×C₈×Dic₁₃	C₄	1	8	yes	complemented
18	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
19	#11718637	—	2	4	no	—
20	#11718637	C₂²	1	8	yes	—
21	C₂×#88048	—	2	8	no	—
22	C₂×#88048	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	—	2	2	no	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂³×Dic₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	D₄	1	4	yes	—
29	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₁₃	—	2	4	no	—
36	C₂×C₄⋊Dic₁₃	—	2	4	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×#6101	—	2	16	no	—
40	C₂×#6101	—	4	16	no	—
41	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
43	C₂×C₄×Dic₁₃	Q₈	1	4	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	4	yes	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₂²×#1055	Q₈	1	4	yes	—
48	C₂²×#1055	C₂³	1	2	yes	—
49	C₁₃×#106	—	2	8	no	—
50	#88057	—	2	8	no	—
51	C₈×Dic₁₃	C₂×C₄	1	16	yes	complemented
52	#88048	—	2	16	no	—
53	C₂×#1844	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	—	4	8	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	4	no	—
67	C₂²×Dic₁₃	—	4	4	no	—
68	C₂²×Dic₁₃	—	4	8	no	—
69	C₂²×Dic₁₃	C₄⋊C₄	1	8	yes	—
70	C₂²×Dic₁₃	SD₁₆	1	16	yes	—
71	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
73	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	4	4	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	4	4	no	—
82	#6101	—	4	32	no	—
83	C₄×Dic₁₃	C₄⋊C₄	1	8	yes	—
84	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
86	C₂×#1055	C₄⋊C₄	1	16	yes	complemented
87	C₂×#1055	C₂²×C₄	1	4	yes	—
88	C₂×#1055	C₂×D₄	1	4	yes	—
89	C₂×#1055	C₂×Q₈	1	4	yes	—
90	C₂²×#106	—	13	1	no	—
91	C₂²×#106	—	13	2	no	—
92	C₂²×#106	D₁₃	1	1	yes	characteristic
93	C₂²×C₄×C₈	—	13	1	no	—
94	C₂×#384	—	13	2	no	—
95	#1844	—	13	8	no	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₁₃	#127	1	2	yes	—
98	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
99	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	4	8	no	—
103	C₂×C₄×C₁₃	—	8	8	no	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	#127	1	2	yes	—
108	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	—	4	8	no	—
119	C₂×Dic₁₃	—	8	8	no	—
120	C₂×Dic₁₃	#106	1	32	yes	complemented
121	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
122	C₂×Dic₁₃	C₂×SD₁₆	1	16	yes	—
123	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
124	#1055	C₂×C₄⋊C₄	1	8	yes	complemented
125	C₂³×C₈	—	13	1	no	—
126	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂²×C₄⋊C₄	—	13	2	no	—
128	C₂²×C₄⋊C₄	—	26	4	no	—
129	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
130	C₂²×C₄²	—	13	1	no	—
131	C₂×#106	—	2	4	no	—
132	C₂×#106	—	13	4	no	—
133	C₂×#106	—	13	8	no	—
134	C₂×#106	—	26	8	no	—
135	C₂×#106	—	26	4	no	—
136	C₂×#106	C₂×D₁₃	1	4	yes	—
137	C₂×C₄×C₈	—	13	8	no	—
138	C₂×C₄×C₈	—	13	4	no	—
139	#384	—	13	8	no	—
140	#384	—	26	4	no	—
141	C₂²×C₁₃	#384	1	4	yes	—
142	C₂²×C₁₃	C₂×#106	1	8	yes	—
143	C₂²×C₁₃	#492	1	8	yes	—
144	C₂²×C₁₃	#498	1	8	yes	—
145	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₁₃	C₂²×SD₁₆	1	4	yes	—
149	C₄×C₁₃	—	8	4	no	—
150	C₄×C₁₃	#384	1	4	yes	—
151	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	Dic₁₃	—	8	4	no	—
156	Dic₁₃	C₂×#106	1	16	yes	complemented
157	C₂³×C₄	—	2	2	no	—
158	C₂³×C₄	—	13	2	no	—
159	C₂³×C₄	—	26	2	no	—
160	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
161	C₂²×C₈	—	13	4	no	—
162	C₂²×C₈	—	13	4	no	—
163	C₂²×C₈	—	13	4	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	Dic₂₆	1	4	yes	—
166	C₂²×C₈	C₄×D₁₃	1	4	yes	—
167	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
168	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	13	4	no	—
173	C₂×C₄⋊C₄	—	26	4	no	—
174	C₂×C₄⋊C₄	—	26	4	no	—
175	C₂×C₄⋊C₄	—	26	16	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	52	16	no	—
178	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
179	C₂×C₄²	—	13	4	no	—
180	C₂×C₄²	—	13	2	no	—
181	C₂×C₄²	—	13	4	no	—
182	C₂×C₄²	—	13	2	no	—
183	#106	—	2	8	no	—
184	#106	—	26	16	no	—
185	#106	—	26	8	no	—
186	C₄×C₈	—	13	16	no	—
187	C₂×C₁₃	#1844	1	8	yes	complemented
188	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
189	C₂×C₁₃	C₂²×#106	1	2	yes	—
190	C₂×C₁₃	C₂×#492	1	2	yes	—
191	C₂×C₁₃	C₂×#498	1	2	yes	—
192	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	13	8	no	—
197	C₂²×C₄	—	13	16	no	—
198	C₂²×C₄	—	13	4	no	—
199	C₂²×C₄	—	26	4	no	—
200	C₂²×C₄	—	52	8	no	—
201	C₂²×C₄	—	52	4	no	—
202	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
204	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₁₃	1	2	yes	—
206	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
207	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
209	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
210	C₂×C₈	—	13	16	no	—
211	C₂×C₈	—	13	4	no	—
212	C₂×C₈	—	13	4	no	—
213	C₂×C₈	—	13	4	no	—
214	C₂×C₈	#6101	1	16	yes	complemented
215	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
216	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
217	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	26	4	no	—
222	C₄⋊C₄	—	26	4	no	—
223	C₄⋊C₄	—	52	32	no	—
224	C₄⋊C₄	—	52	4	no	—
225	C₄²	—	13	8	no	—
226	C₄²	—	13	4	no	—
227	C₄²	—	13	4	no	—
228	C₁₃	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
229	C₂³	#88142	1	2	yes	—
230	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
231	C₂³	SD₁₆×D₁₃	1	8	yes	—
232	C₂³	#88194	1	2	yes	—
233	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	8	no	—
237	C₂×C₄	—	13	8	no	—
238	C₂×C₄	—	13	16	no	—
239	C₂×C₄	—	13	32	no	—
240	C₂×C₄	—	52	8	no	—
241	C₂×C₄	—	52	4	no	—
242	C₂×C₄	—	104	8	no	—
243	C₂×C₄	#88142	1	4	yes	—
244	C₂×C₄	#88142	1	2	yes	—
245	C₂×C₄	C₄⋊C₄×D₁₃	1	4	yes	—
246	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
247	C₂×C₄	C₂×#6101	1	4	yes	—
248	C₂×C₄	#88194	1	2	yes	—
249	C₂×C₄	#88198	1	4	yes	—
250	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₈	—	13	8	no	—
257	C₈	C₂×#6101	1	8	yes	complemented
258	C₂²	#11718516	1	8	yes	—
259	C₂²	D₁₃×#106	1	8	yes	—
260	C₂²	#11718637	1	4	yes	—
261	C₂²	#11718836	1	8	yes	—
262	C₂²	C₂×#88142	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
264	C₂²	C₂×SD₁₆×D₁₃	1	4	yes	—
265	C₂²	C₂×#88194	1	1	yes	characteristic
266	C₄	—	8	4	no	—
267	C₄	—	13	16	no	—
268	C₄	—	104	4	no	—
269	C₄	#11718637	1	4	yes	—
270	C₄	C₂×#88142	1	1	yes	characteristic
271	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
272	C₄	C₂²×#6101	1	1	yes	characteristic
273	C₄	C₂×#88198	1	1	yes	characteristic
274	C₂	#13114991	1	8	yes	direct product factor
275	C₂	C₂×#11718637	1	1	yes	characteristic
276	C₂	C₂×D₁₃×#106	1	2	yes	—
277	C₂	C₂×#11718836	1	2	yes	—
278	C₂	C₂×#11718516	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #13114991	C₂	direct (N × H)
17. C₂×C₈×Dic₁₃	C₄	semidirect
51. C₈×Dic₁₃	C₂×C₄	semidirect
72. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
86. C₂×#1055	C₄⋊C₄	semidirect
120. C₂×Dic₁₃	#106	semidirect
123. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
124. #1055	C₂×C₄⋊C₄	semidirect
156. Dic₁₃	C₂×#106	semidirect
187. C₂×C₁₃	#1844	semidirect
214. C₂×C₈	#6101	semidirect
228. C₁₃	C₂×#1844	semidirect
257. C₈	C₂×#6101	semidirect
274. C₂	#13114991	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#88057	2
C₂²×#88048	2
C₂²×C₈×Dic₁₃	2
C₂×#11718637	2
#13114991	2
C₂²×C₁₃×#106	2
C₂×#1844	13

Subgroups — C2×#13114991 #15863315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114991 #15863315